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Monday, December 2, 2013

Many colleges and universities have a
mathematics or quantitative reasoning requirement that ensures that
no student graduates without completing at least one sufficiently
mathematical course.

Recognizing that taking a regular
first-year mathematics course—designed for students majoring in
mathematics, science, or engineering—to satisfy a QR requirement
is not educationally optimal (and sometimes a distraction for the
instructor and the TAs who have to deal with students who are neither
motivated nor well prepared for the full rigors and pace of a
mathematics course), many institutions offer special QR courses.

I’ve always enjoyed giving such
courses, since they offer the freedom to cover a wide swathe of
mathematics—often new or topical parts of mathematics. Admittedly
they do so at a much more shallow depth than in other courses, but a
depth that was always a challenge for most students who signed up.

Having been one of the pioneers of
so-called “transition courses” for incoming mathematics majors
back in the 1970s, and having given such courses many times in the
intervening years, I never doubted that a lot of the
material was well suited to the student in search of meeting a QR
requirement. The problem with classifying a transition course as a QR
option is that the goal of preparing an incoming student for the
rigors of college algebra and real analysis is at odds with the
intent of a QR requirement. So I never did that.

Enter MOOCs. A lot of the stuff that is
written about these relatively new entrants to the higher education
landscape is unsubstantiated hype and breathless (if not fearful)
speculation. The plain fact is that right now no one really knows
what MOOCs will end up looking like, what part or parts of the
population they will eventually serve, or exactly how and where they
will fit in with the rest of higher education. Like most others I
know who are experimenting with this new medium, I am treating it
very much as just that: an experiment.

The first version of my MOOC
Introduction to Mathematical Thinking, offered in the fall
of 2012, was essentially the first three-quarters of my regular
transition course, modified to make initial entry much easier,
delivered as a MOOC. Since then, as I have experimented with
different aspects of online education, I have been slowly modifying
it to function as a QR-course, since improved quantitative reasoning
is surely a natural (and laudable) goal for online courses with
global reach—that “free education for the world” goal is
still the main MOOC-motivator for me.

I am certainly not viewing my MOOC as
an online course to satisfy a college QR requirement. That may
happen, but, as I noted above, no one has any real idea what role(s)
MOOCs will end up fulfilling. Remember, in just twelve months,
the Stanford MOOC startup Udacity, which initiated all the media
hype, went from “teach the entire world for free” to
“offer corporate training for a fee.” (For my (upbeat) commentary
on this rapid progression, see my article in the Huffington Post.)

Rather, I am taking advantage of the
fact that free, no-credential MOOCs currently provide a superb
vehicle to experiment with ideas both for classroom teaching and for
online education. Those of us at the teaching end not only learn what
the medium can offer, we also discover ways to improve our classroom
teaching; while those who register as students get a totally free
learning opportunity. (Roughly three-quarters of them already have a
college degree, but MOOC enrollees also include thousands of
first-time higher education students from parts of the world that
offer limited or no higher education opportunities.)

The biggest challenge facing anyone who
wants to offer a MOOC in higher mathematics is how to handle the fact
that many of the students will never receive expert feedback on their
work. This is particularly acute when it comes to learning how to
prove things. That’s already a difficult challenge in a regular
class, as made clear in this great blog post by “mathbabe” Cathy O’Neil.
In a MOOC, my current view is it would be unethical to try. The last
thing the world needs are (more) people who think they know
what a proof is, but have never put that knowledge to the test.

But when you think about it, the idea
behind QR is not that people become mathematicians who can prove
things, rather that they have a base level of quantitative literacy
that is necessary to live a fulfilled, rewarding life and be a
productive member of society. Being able to prove something
mathematically is a specialist skill. The important general
ability in today’s world is to have a good understanding of the
nature of the various kinds of arguments, the special nature of
mathematical argument and its role among them, and an ability to
judge the soundness and limitations of any particular argument.

In the case of mathematical argument,
acquiring that “consumer’s understanding” surely
involves having some experience in trying to construct very simple
mathematical arguments, but far more what is required is being able
to evaluate mathematical arguments.

And that can be handled in a MOOC. Just
present students with various mathematical arguments, some correct,
others not, and machine-check if, and how well, they can determine
their validity.

Well, that leading modifier “just”
in that last sentence was perhaps too cavalier. There clearly is more
to it than that. As always, the devil is in the details. But once you
make the shift from viewing the course (or the proofs part of the
course) as being about constructing proofs to being about
understanding and evaluating proofs, then what
previously seemed hopeless suddenly becomes rife with possibilities.

I started to make this shift with the
last session of my MOOC this fall, and though there were significant
teething troubles, I saw enough to be encouraged to try it again—with modifications—to an even greater extent next year.

Of course, many QR courses focus on
appreciation of mathematics, spiced up with enough “doing math”
content to make the course defensibly eligible for QR fulfillment.
What I think is far less common—and certainly new to me—is
using the evaluation of proofs as a major learning vehicle.

What makes this possible is that the
Coursera platform on which my MOOC runs has developed a peer review
module to support peer grading of student papers and exams.

The first times I offered my MOOC, I
used peer evaluation to grade a Final Exam. Though the process worked
tolerably well for grading student mathematics exams—a lot better
than I initially feared—to my eyes it still fell well short of
providing the meaningful grade and expert feedback a professional
mathematician would give. On the other hand, the benefit to the
students that came from seeing, and trying to evaluate, the proof
attempts of other students, and to provide feedback, was significant—both in terms of their gaining much deeper insight into the
concepts and issues involved, and in bolstering their confidence.

When the course runs again in a few
week's time, the Final Exam will be gone, replaced by a new course
culmination activity I am calling Test Flight.

How will it go? I have no idea. That’s
what makes it so interesting. Based on my previous experiments, I
think the main challenges will be largely those of implementation. In
particular, years of educational high-stakes testing robs many
students of the one ingredient essential to real learning: being
willing to take risks and to fail. As young children we have it.
Schools typically drive it out of us. Those of us lucky enough to end
up at graduate school reacquire it—we have to.

I believe MOOCs, which offer community
interaction through the semi-anonymity of the Internet, offer real
potential to provide others with a similar opportunity to re-learn
the power of failure. Test Flight will show if this belief is
sufficiently grounded, or a hopelessly idealistic dream! (Test
flights do sometimes crash and burn.)

The more people learn to view failure
as an essential constituent of good learning, the better life will
become for all. As a world society, we need to relearn that innate
childhood willingness to try and to fail. A society that does not
celebrate the many individual and local failures that are an
inevitable consequence of trying something new, is one destined to
fail globally in the long term.

For those interested, I’ll be
describing Test Flight, and reporting on my progress (including the
inevitable failures), in my blog MOOCtalk.org
as the experiment continues. (The next session starts on February 3.)

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The Mathematical Association of America is the world's largest community of mathematicians, students, and enthusiasts. We accelerate the understanding of our world through mathematics, because mathematics drives society and shapes our lives. Visit us at maa.org.